On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics Convection
This paper is concerned with the nonlinear stability and instability of the two-dimensional (2D) Boussinesq-MHD equations around the equilibrium state <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <m...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/7/1049 |
| _version_ | 1797563830254960640 |
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| author | Dongfen Bian |
| author_facet | Dongfen Bian |
| author_sort | Dongfen Bian |
| collection | DOAJ |
| description | This paper is concerned with the nonlinear stability and instability of the two-dimensional (2D) Boussinesq-MHD equations around the equilibrium state <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mover> <mi>u</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"></mspace> <mover> <mi>B</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"></mspace> <mover> <mi>θ</mi> <mo>¯</mo> </mover> <mo>=</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> with the temperature-dependent fluid viscosity, thermal diffusivity and electrical conductivity in a channel. We prove that if <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mo>+</mo> </msub> <mo>≥</mo> <msub> <mi>a</mi> <mo>−</mo> </msub> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <mfrac> <msup> <mi>d</mi> <mn>2</mn> </msup> <mrow> <mi>d</mi> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mi>κ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> or <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0</mn> <mo><</mo> <mfrac> <msup> <mi>d</mi> <mn>2</mn> </msup> <mrow> <mi>d</mi> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mi>κ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mi>β</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula>, with <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>β</mi> <mn>0</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> small enough constant, and then this equilibrium state is nonlinearly asymptotically stable, and if <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mo>+</mo> </msub> <mo><</mo> <msub> <mi>a</mi> <mo>−</mo> </msub> </mrow> </semantics> </math> </inline-formula>, this equilibrium state is nonlinearly unstable. Here, <inline-formula> <math display="inline"> <semantics> <msub> <mi>a</mi> <mo>+</mo> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>a</mi> <mo>−</mo> </msub> </semantics> </math> </inline-formula> are the values of the equilibrium temperature <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> on the upper and lower boundary. |
| first_indexed | 2024-03-10T18:47:47Z |
| format | Article |
| id | doaj.art-c6f891cd4b45416399d2b63360731b0f |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T18:47:47Z |
| publishDate | 2020-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c6f891cd4b45416399d2b63360731b0f2023-11-20T05:22:29ZengMDPI AGMathematics2227-73902020-06-0187104910.3390/math8071049On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics ConvectionDongfen Bian0School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, ChinaThis paper is concerned with the nonlinear stability and instability of the two-dimensional (2D) Boussinesq-MHD equations around the equilibrium state <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mover> <mi>u</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"></mspace> <mover> <mi>B</mi> <mo>¯</mo> </mover> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"></mspace> <mover> <mi>θ</mi> <mo>¯</mo> </mover> <mo>=</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> with the temperature-dependent fluid viscosity, thermal diffusivity and electrical conductivity in a channel. We prove that if <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mo>+</mo> </msub> <mo>≥</mo> <msub> <mi>a</mi> <mo>−</mo> </msub> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <mfrac> <msup> <mi>d</mi> <mn>2</mn> </msup> <mrow> <mi>d</mi> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mi>κ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> or <inline-formula> <math display="inline"> <semantics> <mrow> <mn>0</mn> <mo><</mo> <mfrac> <msup> <mi>d</mi> <mn>2</mn> </msup> <mrow> <mi>d</mi> <msup> <mi>y</mi> <mn>2</mn> </msup> </mrow> </mfrac> <mi>κ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mi>β</mi> <mn>0</mn> </msub> </mrow> </semantics> </math> </inline-formula>, with <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>β</mi> <mn>0</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> small enough constant, and then this equilibrium state is nonlinearly asymptotically stable, and if <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mo>+</mo> </msub> <mo><</mo> <msub> <mi>a</mi> <mo>−</mo> </msub> </mrow> </semantics> </math> </inline-formula>, this equilibrium state is nonlinearly unstable. Here, <inline-formula> <math display="inline"> <semantics> <msub> <mi>a</mi> <mo>+</mo> </msub> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <msub> <mi>a</mi> <mo>−</mo> </msub> </semantics> </math> </inline-formula> are the values of the equilibrium temperature <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>θ</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> on the upper and lower boundary.https://www.mdpi.com/2227-7390/8/7/1049Boussinesq-MHD systemasymptotic stabilitynonlinear instability |
| spellingShingle | Dongfen Bian On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics Convection Mathematics Boussinesq-MHD system asymptotic stability nonlinear instability |
| title | On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics Convection |
| title_full | On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics Convection |
| title_fullStr | On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics Convection |
| title_full_unstemmed | On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics Convection |
| title_short | On the Nonlinear Stability and Instability of the Boussinesq System for Magnetohydrodynamics Convection |
| title_sort | on the nonlinear stability and instability of the boussinesq system for magnetohydrodynamics convection |
| topic | Boussinesq-MHD system asymptotic stability nonlinear instability |
| url | https://www.mdpi.com/2227-7390/8/7/1049 |
| work_keys_str_mv | AT dongfenbian onthenonlinearstabilityandinstabilityoftheboussinesqsystemformagnetohydrodynamicsconvection |